Understanding Compound Interest and Half-Yearly Compounding

Compounding interest is one of the fundamental concepts in finance and mathematics. It refers to the process of earning interest on both the initial principal and the accumulated interest over a specified period. When interest is compounded half-yearly, it means the interest is calculated and added to the principal every six months. This tutorial explains how to calculate the compound interest rate for a given principal amount, time period, and final amount.

Problem Statement and Solution

Given the principal amount, the final amount, and the time period, we need to find the rate of interest per annum when compounded half-yearly. This example will guide you through the process using a specific scenario.

Step-by-Step Calculation

The principal amount (P) is Rs 2000, the final amount (A) is Rs 2662, and the time period (n) is 3/2 years, which is equivalent to 3 half-years.

Using the Compound Interest Formula

The compound interest formula is given by:

A P (1 in)

Where:

A is the final amount. P is the principal amount. i is the interest rate per compounding period. n is the number of compounding periods.

Substituting the given values:

2662 2000 (1 i3)

(1 i3) 2662/2000 1.331

Calculating the Interest Rate Per Half-Yearly Period

To find i3:

1 i3 1.331

i3 1.331 - 1 0.331

Now, solving for i:

1 i (1.331)1/3

1 i 1.101

i 1.101 - 1 0.101 or 10.1%

Since the interest rate per half-yearly period is 10.1%, the annual interest rate is:

R 10.1% × 2 20.2%

Rounding off, the required rate is 20% per annum.

Alternative Method for Verification

There is a second approach to verify the result using the formula directly without cubing:

A P (1 in)

1852.50 1600 (1 i3)

(1 i3) 1852.50/1600 1.1578

1 i (1.1578)1/3

1 i 1.05

i 1.05 - 1 0.05 or 5%

Since the interest rate per half-yearly period is 5%, the annual interest rate is:

R 5% × 2 10%

This verification shows that the rate per annum is 10% when compounded half-yearly.

Conclusion

Understanding and calculating compound interest rates is crucial for financial planning and investment management. Whether you are dealing with annual or half-yearly compounding, mastering the calculation process ensures accurate financial predictions and better investment decisions.